Certainly! Let's analyze the equation [tex]\( y = -7x + 4 \)[/tex] step by step:
1. Identifying the Type of Equation:
- The given equation is of the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- Since this is a straight-line equation, it represents a linear function.
2. Determining if it Represents a Function:
- For every value of [tex]\( x \)[/tex], there is exactly one corresponding value of [tex]\( y \)[/tex]. Therefore, it satisfies the definition of a function.
3. Providing Three Coordinates:
- To find three coordinates, we can choose three different values for [tex]\( x \)[/tex] and calculate the corresponding [tex]\( y \)[/tex] values.
- Let's pick [tex]\( x \)[/tex] values of [tex]\( 0 \)[/tex], [tex]\( 1 \)[/tex], and [tex]\( 2 \)[/tex]:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[
y = -7(0) + 4 = 4
\][/tex]
The coordinate is [tex]\( (0, 4) \)[/tex].
- When [tex]\( x = 1 \)[/tex]:
[tex]\[
y = -7(1) + 4 = -3
\][/tex]
The coordinate is [tex]\( (1, -3) \)[/tex].
- When [tex]\( x = 2 \)[/tex]:
[tex]\[
y = -7(2) + 4 = -10
\][/tex]
The coordinate is [tex]\( (2, -10) \)[/tex].
4. Describing the Pattern:
- Observing the coordinates, we see the values [tex]\( (0, 4) \)[/tex], [tex]\( (1, -3) \)[/tex], [tex]\( (2, -10) \)[/tex].
- We notice that as [tex]\( x \)[/tex] increases by [tex]\( 1 \)[/tex], [tex]\( y \)[/tex] decreases by [tex]\( 7 \)[/tex]. This is consistent with the slope [tex]\( m = -7 \)[/tex] of the linear equation.
Conclusion:
- The equation [tex]\( y = -7x + 4 \)[/tex] represents a linear function.
- Three coordinates for this linear function are [tex]\( (0, 4) \)[/tex], [tex]\( (1, -3) \)[/tex], and [tex]\( (2, -10) \)[/tex].
- The pattern is that [tex]\( y \)[/tex] decreases by [tex]\( 7 \)[/tex] units for every [tex]\( 1 \)[/tex] unit increase in [tex]\( x \)[/tex].
